Barnett, a redshirt sophomore, left Alabama on Sept. 28, four games into the 2016 season, after freshman Jalen Hurts became established as the Crimson Tide's starter. As a transfer, he'd typically be required to sit out a calendar year if not for the ruling, which was first reported by The Arizona Republic.
Instead, he will compete with returning starter Manny Wilkins, a junior, to run the Sun Devils' offense in the season opener on Aug. 31 against New Mexico State.
Barnett shared his excitement on Twitter after the ruling.
Arizona State also this week announced the hiring of Billy Napier as offensive coordinator and quarterbacks coach. Napier was Alabama's receivers coach, so he is familiar with Barnett.
Barnett, who played at Santiago High School in Corona, California, was rated the No. 1 pocket passer in the 2015 recruiting class. He started Alabama's 2016 season opener against Southern California but lost his job to Hurts. He played in three of the first four games, completing 11 of 19 passes for 219 yards with two touchdowns and no interceptions.
Barnett enrolled in Palomar College, a community college, after leaving Tuscaloosa. That allowed him to become eligible about a month into the 2017 season, but he appealed to the NCAA for the right to play the whole season.
Injuries at quarterback helped derail last season for the Sun Devils (5-7). Wilkins finished the season as the starter. Brady White and Bryce Perkins are both still recovering from injuries and might not be ready for spring practice. The Sun Devils also have Dillon Sterling-Cole, who played in six games last season, and they expect to sign four-star prospect Ryan Kelley on Wednesday.
The Associated Press contributed to this report.